Question 1

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard.
A) The z-statistic is -2.25
B) The z-statistic is 2.25
C) The z-statistic is -1.00
D) The z-statistic is 1.00

Accepted Answer

To determine the z-statistic, we can use the formula:
z = (x - μ) / (σ / sqrt(n))
where x is the sample proportion (10/500 = 0.02), μ is the population proportion (0.01), σ is the standard deviation (sqrt(0.01 * 0.99) = 0.0995), and n is the sample size (500).

Plugging in the values, we get:
z = (0.02 - 0.01) / (0.0995 / sqrt(500)) = 2.24

Rounding to two decimal places, we get a z-statistic of 2.25, which corresponds to answer choice B. This result indicates that the sample proportion of products that did not meet the standard is more than 2 standard deviations away from the population proportion, suggesting that the manufacturer's claim is unlikely to be true.

Question 2

i. If the critical values of the test statistic z are  $\pm$1.96, they are the dividing points between the areas of rejection and non-rejection. ii. The probability of a Type I error is also referred to as alpha.
iii. A Type I error is the probability of accepting a true null hypothesis.
A) (i), (ii), and (iii) are all correct statements.
B) (i) is a correct statement but not (ii) or (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (i) and (ii) are correct statements but not (iii).
E) (i), (ii), and (iii) are all false statements.

Accepted Answer

Statement (i) is correct as the critical values of the test statistic z do indeed separate the areas of rejection and non-rejection. Statement (ii) is also correct as the probability of a Type I error is commonly denoted as alpha. However, statement (iii) is incorrect as a Type I error is the probability of rejecting a true null hypothesis, not accepting it.

Question 3

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard.
A) If $\alpha$ = .01, the critical value is z = -2.33
B) If $\alpha$ = .01, the critical value is z = 2.33
C) If $\alpha$ = .01, the critical value is z = -1.96
D) If $\alpha$ = .01, the critical value is z = 1.96
E) If $\alpha$ = .01, the critical value is z = -2.58

Accepted Answer

The answer of A manufacturer claims that less than 1%...

Question 4

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the null hypothesis?
A) p $\neq$ 0.10
B) p $\ge$ 0.10
C) p $\le$ 0.10
D) p < 0.10
E) p = 0.10

Accepted Answer

The null hypothesis is that the proportion of defective peaches is at least 10%, i.e., p ≥ 0.10.

Question 5

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard. A) The null hypothesis is p $\ge$ 0.01. The alternate hypothesis is p < 0.01 B) The null hypothesis is p = 0.01. The alternate hypothesis is p > 0.01 C) The null hypothesis is p $\ge$ 0.01. The alternate hypothesis is p > 0.01 D) The null hypothesis is p = 0.01. The alternate hypothesis is p $\neq$ 0.01 E) The null hypothesis is p < 0.01. The alternate hypothesis is p $\neq$ 0.01

Accepted Answer

The null hypothesis should reflect the manufacturer's claim that the proportion of products not meeting standards is at least 1% (p ≥ 0.01), while the alternative hypothesis should reflect the claim being tested, which is that the proportion is less than 1% (p < 0.01).

Question 6

If $\alpha$ = 0.05, what is the probability of making a Type I error?
A) 0
B) 1/20
C) 19/20
D) 20/20

Accepted Answer

F1F1F1S1F1F1F10 = 0.05 represents the significance level (alpha) of the hypothesis test. The probability of making a Type I error is equal to the significance level, which is 0.05 or 1/20. Therefore, the best choice is B.

Question 7

i. The probability of a Type I error is also referred to as alpha. ii. A Type I error is the probability of accepting a true null hypothesis.
iii. A Type I error is the probability of rejecting a true null hypothesis.
A) (i), (ii), and (iii) are all correct statements.
B) (i) is a correct statement but not (ii) or (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (i) and (ii) are correct statements but not (iii).
E) (i), (ii), and (iii) are all false statements.

Accepted Answer

The answer of i. The probability of a Type I...

Question 8

What is the probability of making a Type II error if the null hypothesis is actually true?
A) $\alpha$
B) 1
C) 0
D) 0.05

Accepted Answer

The probability of making a Type II error is the probability of accepting the null hypothesis when it is actually false. If the null hypothesis is true, then there is no chance of making a Type II error, as the null hypothesis being true means that the alternative hypothesis is false. Therefore, the answer is 0.

Question 9

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the sample proportion?
A) 0.10
B) 0.125
C) 40
D) 0.40

Accepted Answer

The sample proportion is calculated by dividing the number of defective peaches (50) by the total number of peaches examined (400), which equals 0.125.

Question 10

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard. If the z-statistic is -1.96 and the level of significance is 0.01, what is your decision?
A) You reject the null hypothesis.
B) You have insufficient evidence to reject the null hypothesis.
C) You accept the alternate hypothesis and agree that less than 1% meet government standards.
D) You should have used the t-test.

Accepted Answer

The null hypothesis is that the proportion of products not meeting the standard is equal to or greater than 1%, and the alternate hypothesis is that the proportion is less than 1%. We can use a one-tailed z-test for proportions to test this.

The calculated z-statistic (-1.96) falls in the rejection region (left tail) at a level of significance of 0.01. However, we need to compare this to the critical value of the z-distribution, which is -2.33 for a one-tailed test at a 0.01 level of significance with a sample size of 500. Since our calculated z-statistic (-1.96) is greater than the critical value (-2.33), we have insufficient evidence to reject the null hypothesis. Therefore, we conclude that there is no significant evidence to suggest that the proportion of products not meeting the standard is less than 1%.

Question 11

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard. If the level of significance is 2%, what is the critical value?
A) 2.05
B) -2.05
C) 2.33
D) -2.33
E) 1.96

Accepted Answer

The critical value for a 2% level of significance in a two-tailed test corresponds to a Z-score of approximately ±2.05. However, since the claim is "less than 1%", we are dealing with a one-tailed test, and the negative sign indicates we are looking at the left tail of the distribution, which aligns with the idea of finding values less than a certain threshold.

Question 12

Based on the Nielsen ratings, the local CBS affiliate claims its 11:00 PM newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers, 36% indicated that they watch the late evening news on this local CBS station. What is the critical value if the level of significance is 0.10?
A) -1.282
B)  $\pm$ 1.65
C) -2.58
D) 2.58

Accepted Answer

The answer of Based on the Nielsen ratings, the local...

Question 13

What is the level of significance?
A) Probability of a Type II error
B) Probability of a Type I error
C) z-value of 1.96
D) Beta error

Accepted Answer

The level of significance is the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. This is the most commonly used definition of the level of significance in hypothesis testing.

Question 14

Based on the Nielsen ratings, the local CBS affiliate claims its 11:00 PM newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers, 36% indicated that they watch the late evening news on this local CBS station. What is your decision if $\alpha$ = 0.01?
A) Fail to reject the null hypothesis and conclude about 41%.
B) Reject the null hypothesis and conclude different from 41%.
C) Fail to reject the alternate and conclude different from 41%.
D) Reject the alternate and conclude it is 41%.

Accepted Answer

The sample proportion (36%) is close to the population proportion claimed by the CBS affiliate (41%), and without further statistical tests provided (like a z-test or p-value), we cannot conclude that there is a significant difference. Therefore, we fail to reject the null hypothesis, which would state that the true proportion is about 41%.

Question 15

The mean gross annual incomes of certified tack welders are normally distributed with the mean of $50,000 and a standard deviation of $4,000. The ship building association wishes to find out whether their tack welders earn more or less than $50,000 annually. The alternate hypothesis is that the mean is not $50,000. When tested at the 99% level, these are the results found with megastat: 
A) Since the p-value is 0.43 at this level of testing, there is insufficient evidence to say that the welders earn an income different from $50,000.
B) Since the p-value is 0.43 at this level of testing, there is sufficient evidence to say that the welders earn more than $50,000.
C) The p-value of 0.043 indicates that there is sufficient evidence to say that the welders earn more than $50,000.

Accepted Answer

At the 99% level of significance, the p-value (0.43) is greater than the level of significance (0.01). This indicates that we fail to reject the null hypothesis and do not have sufficient evidence to say that the mean income of tack welders is different from $50,000. Therefore, option A is correct. Option B is incorrect because the p-value does not support the claim that welders earn more than $50,000. Option C is also incorrect because the p-value is greater than the level of significance, and we cannot conclude that the welders earn more than $50,000.

Question 16

What is a Type II error?
A) Accepting a false null hypothesis
B) Rejecting a false null hypothesis
C) Accepting a false alternate hypothesis
D) Rejecting a false alternate hypothesis

Accepted Answer

A Type II error occurs when we fail to reject a false null hypothesis, meaning that we incorrectly accept the null hypothesis when it is actually false. This is also known as a false negative.

Question 17

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the z-statistic?
A) 0.025
B) 0.278
C) -1.65
D) 1.67

Accepted Answer

The z-statistic is calculated using the formula $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and $n$ is the sample size. Here, $\hat{p} = \frac{50}{400} = 0.125$, $p_0 = 0.10$, and $n = 400$. Plugging these values into the formula gives $z = \frac{0.125 - 0.10}{\sqrt{\frac{0.10(1-0.10)}{400}}} = 1.67$.

Question 18

i. If the critical values of the test statistic z are  $\pm$1.96, they are the dividing points between the areas of rejection and non-rejection. ii. The probability of a Type I error is also referred to as alpha.
iii. A Type I error is the probability of rejecting a true null hypothesis.
A) (i), (ii), and (iii) are all correct statements.
B) (i) is a correct statement but not (ii) or (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (i) and (ii) are correct statements but not (iii).
E) (i), (ii), and (iii) are all false statements.

Accepted Answer

The critical values of the test statistic are indeed the dividing points between rejection and non-rejection areas. The probability of a Type I error is referred to as alpha, and a Type I error occurs when a true null hypothesis is rejected.

Question 19

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the critical value for $\alpha$ = 0.025?
A) 1.96
B)  $\pm$1.65
C) -1.96
D) -1.65

Accepted Answer

The answer of It is claimed that in a bushel...

Question 20

The mean gross annual incomes of certified tack welders are normally distributed with the mean of $50,000 and a standard deviation of $4,000. The ship building association wishes to find out whether their tack welders earn more or less than $50,000 annually. The alternate hypothesis is that the mean is not $50,000. Given the following megastat printout, what conclusions can be made?

A) There is a 1.68% chance of getting these sample results if the annual income is actually $50,000.
B) Since the p-value is 7%, we reject the null hypothesis at a 5% level of significance.
C) Since the p-value is 9%, we have insufficient evidence to reject the null hypothesis at the 5% level of significance.
D) Since the p-value is 9%, we reject the null hypothesis at the 1% level of significance.
E) Since the p-value is incredibly small (far less than 1%), we reject the null hypothesis at the 1% level of significance.

Accepted Answer

The answer of The mean gross annual incomes of certified...

Quiz 9: One-Sample Tests of Hypothesis

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard.

i. If the critical values of the test statistic z are $\pm$ 1.96, they are the dividing points between the areas of rejection and non-rejection. ii. The probability of a Type I error is also referred to as alpha. iii. A Type I error is the probability of accepting a true null hypothesis.

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard.

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the null hypothesis?

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard.

If $\alpha$ = 0.05, what is the probability of making a Type I error?

i. The probability of a Type I error is also referred to as alpha. ii. A Type I error is the probability of accepting a true null hypothesis. iii. A Type I error is the probability of rejecting a true null hypothesis.

What is the probability of making a Type II error if the null hypothesis is actually true?

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the sample proportion?

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard. If the z-statistic is -1.96 and the level of significance is 0.01, what is your decision?

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard. If the level of significance is 2%, what is the critical value?

What is the level of significance?

Based on the Nielsen ratings, the local CBS affiliate claims its 11:00 PM newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers, 36% indicated that they watch the late evening news on this local CBS station. What is your decision if $\alpha$ = 0.01?

What is a Type II error?

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the z-statistic?

i. If the critical values of the test statistic z are $\pm$ 1.96, they are the dividing points between the areas of rejection and non-rejection. ii. The probability of a Type I error is also referred to as alpha. iii. A Type I error is the probability of rejecting a true null hypothesis.

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the critical value for $\alpha$ = 0.025?

Quiz 9: One-Sample Tests of Hypothesis

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard.

i. If the critical values of the test statistic z are ±\pm± 1.96, they are the dividing points between the areas of rejection and non-rejection. ii. The probability of a Type I error is also referred to as alpha. iii. A Type I error is the probability of accepting a true null hypothesis.

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard.

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the null hypothesis?

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard.

If α\alphaα = 0.05, what is the probability of making a Type I error?

i. The probability of a Type I error is also referred to as alpha. ii. A Type I error is the probability of accepting a true null hypothesis. iii. A Type I error is the probability of rejecting a true null hypothesis.

What is the probability of making a Type II error if the null hypothesis is actually true?

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the sample proportion?

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard. If the z-statistic is -1.96 and the level of significance is 0.01, what is your decision?

A manufacturer claims that less than 1% of all his products do not meet the minimum government standards. A survey of 500 products revealed ten did not meet the standard. If the level of significance is 2%, what is the critical value?

What is the level of significance?

Based on the Nielsen ratings, the local CBS affiliate claims its 11:00 PM newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers, 36% indicated that they watch the late evening news on this local CBS station. What is your decision if α\alphaα = 0.01?

What is a Type II error?

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the z-statistic?

i. If the critical values of the test statistic z are ±\pm± 1.96, they are the dividing points between the areas of rejection and non-rejection. ii. The probability of a Type I error is also referred to as alpha. iii. A Type I error is the probability of rejecting a true null hypothesis.

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the critical value for α\alphaα = 0.025?

i. If the critical values of the test statistic z are $\pm$ 1.96, they are the dividing points between the areas of rejection and non-rejection. ii. The probability of a Type I error is also referred to as alpha. iii. A Type I error is the probability of accepting a true null hypothesis.

If $\alpha$ = 0.05, what is the probability of making a Type I error?

Based on the Nielsen ratings, the local CBS affiliate claims its 11:00 PM newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers, 36% indicated that they watch the late evening news on this local CBS station. What is your decision if $\alpha$ = 0.01?

i. If the critical values of the test statistic z are $\pm$ 1.96, they are the dividing points between the areas of rejection and non-rejection. ii. The probability of a Type I error is also referred to as alpha. iii. A Type I error is the probability of rejecting a true null hypothesis.

It is claimed that in a bushel of peaches less than ten percent are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the critical value for $\alpha$ = 0.025?